Mathematics
Master of Science: 45.0 quarter credits
Doctor of Philosophy: 90.0 quarter credits
About the Program
The Department of Mathematics is a broadly based academic unit offering instructional programs and carrying on research activities in mathematics. Doctor of Philosophy and Master of Science degrees are offered.
Areas of research specialty among the faculty include applied mathematics, algebraic combinatorics, biomathematics, discrete mathematics, optics, analysis, number theory, numerical analysis, probability and statistics, matrix and operator theory, fluid mechanics, and partial differential equations.
Additional Information
For more information about theses graduate programs, visit Drexel University's Mathematics webpage.
Admission Requirements
Applicants should hold a BS degree in mathematics or the equivalent and meet the University's graduate admission standards. In particular, the student should have had intensive exposure to proof oriented courses, such as real analysis and abstract algebra. Students requesting financial aid are required to take the Graduate Record Examination General Test. Because many of the core courses are two- or three-term sequences beginning in the fall, new students are typically admitted to the programs only in the fall term. Admissions standards for the MS and PhD programs are equivalent.
For additional information on how to apply, visit Drexel University's Graduate Admissions website.
Master of Science in Mathematics
Students must complete a minimum of 45.0 graduate credits for the MS degree. Of these 15 courses, the following six are required:
| Required Courses | ||
| MATH 504 | Linear Algebra & Matrix Analysis | 3.0 |
| MATH 505 | Principles of Analysis I | 3.0 |
| MATH 506 | Principles of Analysis II | 3.0 |
| MATH 533 | Abstract Algebra I | 3.0 |
| MATH 630 | Complex Variables I | 3.0 |
| MATH 633 | Real Variables I | 3.0 |
The remaining 9 courses may be any graduate mathematics courses. In some cases, course substitutions may be made with courses from other departments. Elective courses taken outside the department must receive prior departmental approval in order to be counted toward the degree.
There are no thesis, language, or special examination requirements for the master's degree.
Students seeking a dual MS must satisfy core requirements for both degree programs.
Students should note that some departmental courses, such as Advanced Engineering Mathematics, are foundation courses and do not contribute to the departmental requirements for the degree. They do count toward the University requirements for a degree.
PhD in Mathematics
Students must complete a minimum of 45 graduate credits for the PhD degree, in addition to the 45.0 required by the MS program for a total of 90.0 credits. Of the 45.0 credits of MS program courses, the following six are required:
| Required Courses | ||
| MATH 504 | Linear Algebra & Matrix Analysis | 3.0 |
| MATH 505 | Principles of Analysis I | 3.0 |
| MATH 506 | Principles of Analysis II | 3.0 |
| MATH 533 | Abstract Algebra I | 3.0 |
| MATH 630 | Complex Variables I | 3.0 |
| MATH 633 | Real Variables I | 3.0 |
The remaining 27.0 credits, comprising the MS segment of the PhD program, may be any graduate mathematics courses. In some cases, course substitutions may be made with courses from other departments. Elective courses taken outside the department must receive prior departmental approval in order to be counted toward the degree.
The student must pass a written qualifying exam. The student is allowed two attempts. Students must take exam at the end of their first year, and have a second opportunity in September of their second year.
Students must take a PhD candidacy exam at the end of their second year. Additional coursework to reach the 90.0 credits required for the PhD will be agreed upon with the student's Graduate Advisor. Students should note that some departmental courses, such as MATH 544 Advanced Engineering Mathematics, are foundation courses and do not contribute to the departmental requirements for the degree. They do count toward the University requirements for a degree.
Facilities
Department computers are accessible from residence halls over the campus network, and from off-campus via modem or an Internet Service Provider (ISP). Departmental and university networks provide access to the Internet and the Pennsylvania Education Network (PrepNET). Departmental research computers have a connection to the campus backbone at 100 Mbps and are also on the vBNS via a campus OCS ATM connection.
The computing resources of the Mathematics Department include:
- Math Resource Center (Korman 247): 6 Dell Optiplex (Core 2 Duo 2.8 Ghz, 3 GB RAM) running Windows XP Professional SP3.
- Faculty Center (Korman 207): 2 Lenovo ThinkCentre (Pentium 4 3.0 Ghz, 1 GB RAM) running Windows XP Professional SP3.
- Computer Server: One Penguin Server (Dual 2.2. GHz Opteron, 8 GB RAM) running Ubuntu Linux.
- File/Print/Mail/Web Server: 2 Penguin Servers (Dual 2.8 GHz Zeon, 1 GB RAM) running Ubuntu Linux and connected to 600GB RAID 5 Disk over a fully switched gigabit Ethernet network, 2TB mirrored RAID.
Courses
MATH 504 Linear Algebra & Matrix Analysis 3.0 Credits
Course topics include the QR decomposition, Schur's triangularization theorem, the spectral decomposition for normal matrices, the Jordan canonical form, the Courant-Fisher theorem, singular value and polar decompositions, the Gersgorin disc theorem, the Perron-Frobenius theorem, and other current matrix analysis topics. Applications of the material are outlined as well.
Repeat Status: Not repeatable for credit
MATH 505 Principles of Analysis I 3.0 Credits
Metric spaces, compactness, connectedness, completeness. Set theory and cardinality, continuity, differentiation, Riemann integral.
Repeat Status: Not repeatable for credit
MATH 506 Principles of Analysis II 3.0 Credits
A continuation of MATH 505. Uniform convergence, Fourier series, Lebesque integral in Euclidean spaces, differential calculus in Euclidean spaces, inverse and implicit functions theorems, change of variables in multiple integrals.
Repeat Status: Not repeatable for credit
Prerequisites: MATH 505 [Min Grade: C]
MATH 507 Applied Mathematics I 3.0 Credits
Covers matrix theory, linear transformations, canonical forms, matrix decompositions, and factorizations, including the singular value decomposition, quadratic forms, matrix least squares problems, and fast unitary transforms. Introduces computational linear algebra.
Repeat Status: Not repeatable for credit
MATH 508 Applied Mathematics II 3.0 Credits
Covers the techniques of mathematical modeling in the physical and biological sciences using discrete and combinatorial mathematics, probabilistic methods, variational principles, Fourier series and integrals, integral equations, calculus of variations, asymptotic series and expansions, and eigenvalue problems associated with Sturm-Liouville boundary value problems. Topics vary from year to year.
Repeat Status: Not repeatable for credit
Prerequisites: MATH 507 [Min Grade: C]
MATH 509 Applied Mathematics III 3.0 Credits
Continues the theme of MATH 508. Topics vary from year to year.
Repeat Status: Not repeatable for credit
Prerequisites: MATH 508 [Min Grade: C]
MATH 510 Applied Probability and Statistics I 3.0 Credits
Covers basic concepts in applied probability; random variables, distribution functions, expectations, and moment generating functions; specific continuous and discrete distributions and their properties; joint and conditional distributions; discrete time Markov chains; distributions of functions of random variables; probability integral transform; and central limit theorem.
Repeat Status: Not repeatable for credit
MATH 511 Applied Probability and Statistics II 3.0 Credits
Covers probability plots and graphical techniques for determining distribution of data, including sampling and sampling distributions, law of large numbers, parametric point estimation, maximum likelihood estimation, Bayes estimation, properties of estimators, sufficient statistics, minimum variance unbiased estimators, and parametric interval estimation. Introduces hypothesis testing.
Repeat Status: Not repeatable for credit
Prerequisites: MATH 510 [Min Grade: C]
MATH 512 Applied Probability and Statistics III 3.0 Credits
Covers hypothesis testing, analysis of variance, multiple regression, and special topics. Introduces linear models.
Repeat Status: Not repeatable for credit
Prerequisites: MATH 511 [Min Grade: C]
MATH 520 Numerical Analysis I 3.0 Credits
Covers polynomial interpolation, numerical solutions of nonlinear equations, numerical integration (Newton-Cotes, Gauss quadrature), error estimates of various numerical methods, and function approximation (polynomial, Fourier, Pade).
Repeat Status: Not repeatable for credit
MATH 521 Numerical Analysis II 3.0 Credits
Covers numerical linear algebra and matrix computation, direct and iterative methods for solving linear systems and eigenvalue problems, least square problems, various matrix factorizations (QR, singular value decomposition, LU and Cholesky), and Krylov subspace methods.
Repeat Status: Not repeatable for credit
MATH 522 Numerical Analysis III 3.0 Credits
Covers numerical solutions of ordinary and partial differential equations.
Repeat Status: Not repeatable for credit
Prerequisites: MATH 520 [Min Grade: C]
MATH 523 Computer Simulation I 3.0 Credits
Covers computer simulation of pseudo-random variables, including Monte Carlo methods.
Repeat Status: Not repeatable for credit
Prerequisites: MATH 510 [Min Grade: C]
MATH 524 Computer Simulation II 3.0 Credits
Covers discrete and continuous event simulation models and techniques.
Repeat Status: Not repeatable for credit
Prerequisites: MATH 523 [Min Grade: C]
MATH 525 Topics in Computer Simulation 3.0 Credits
Covers statistical analysis of simulation data, variance reduction techniques, and advanced topics in simulation.
Repeat Status: Not repeatable for credit
Prerequisites: MATH 524 [Min Grade: C]
MATH 530 Combinatorial Mathematics I 3.0 Credits
Covers graphs and networks, with an emphasis on algorithms. Includes minimum spanning trees, shortest path problems, connectivity, network flows, matching theory, Eulerian and Hamiltonian tours, graph coloring, and random graphs.
Repeat Status: Not repeatable for credit
MATH 531 Combinatorial Mathematics II 3.0 Credits
Covers mathematical tools for the analysis of algorithms, including combinatorics, recurrence relations and generating functions, elementary asymptotics, and probabilistic methods.
Repeat Status: Not repeatable for credit
Prerequisites: MATH 530 [Min Grade: C]
MATH 532 Topics in Combinatorial Math 3.0 Credits
Covers topics in discrete mathematics, including asymptotic enumeration, number theory, probabilistic combinatorics, and combinatoric algorithms.
Repeat Status: Not repeatable for credit
Prerequisites: MATH 531 [Min Grade: C]
MATH 533 Abstract Algebra I 3.0 Credits
Covers groups, transformation groups and group actions, isomorphism and homomorphism theorems, Sylow theorems, symmetric groups, rings, and fields.
Repeat Status: Not repeatable for credit
MATH 534 Abstract Algebra II 3.0 Credits
Covers factorization domains, Euclidean domains, and polynomial rings, and modules, vector spaces, and linear transformations.
Repeat Status: Not repeatable for credit
Prerequisites: MATH 533 [Min Grade: C]
MATH 535 Topics in Abstract Algebra 3.0 Credits
This third course in the Abstract Algebra sequence covers a selection of topics in advanced modern algebra such as symmetries, representation theory, algebraic geometry, homological algebra, Galois Theory and coding theory.
Repeat Status: Can be repeated 3 times for 9 credits
Prerequisites: MATH 533 [Min Grade: C] and MATH 534 [Min Grade: C]
MATH 536 Topology I 3.0 Credits
Covers general topological spaces, metric spaces, and function spaces; open sets, limit points, limits of sequences, convergence, separation axioms, compactness, connectedness, continuity, homeomorphisms, and product of N-spaces; and specialized applications to the real line, Euclidean N-space, and well-known function spaces.
Repeat Status: Not repeatable for credit
MATH 537 Topology II 3.0 Credits
Continues MATH 536.
Repeat Status: Not repeatable for credit
MATH 538 Manifolds 3.0 Credits
Topics will be selected from the following: Differential structures, immersion theorems, tangent bundles, vector fields and distributions, integral manifolds, integration on manifolds, differential forms, general Stokes Theorem, applications to physics and engineering.
Repeat Status: Not repeatable for credit
MATH 540 Numerical Computing 3.0 Credits
Intended to introduce students to contemporary computing environments and the associated tools. Uses contemporary software tools and specific applications from science and engineering to illustrate numerical and visualization methods.
Repeat Status: Not repeatable for credit
MATH 544 Advanced Engineering Mathematics I 3.0 Credits
Covers solution techniques for ordinary differential equations, including series techniques, Legendre and Bessel functions, Sturm-Liouville theory, and Laplace and Fourier techniques. Introduces symbolic computation as time permits.
Repeat Status: Not repeatable for credit
MATH 545 Advanced Engineering Mathematics II 3.0 Credits
Covers partial differential equations, including separation of variables and its applications to standard equations. Introduces Green's functions for differential equations.
Repeat Status: Not repeatable for credit
Prerequisites: MATH 544 [Min Grade: C]
MATH 546 Advanced Engineering Mathematics III 3.0 Credits
Covers complex analysis, including complex differentiation and integration, Cauchy's theorems and residue theory, and their applications; conformal maps; and applications to fluid flow.
Repeat Status: Not repeatable for credit
Prerequisites: MATH 545 [Min Grade: C]
MATH 553 Sci Comp & Visualization I 3.0 Credits
Covers scientific computing, with an emphasis on numerical computing and visualization techniques. Includes techniques of computational geometry, including an introduction to methods used to describe the shapes of free-form curves, surfaces, and volumes, and applications to computer-aided design and other areas of scientific computing.
Repeat Status: Not repeatable for credit
Prerequisites: MATH 540 [Min Grade: C]
MATH 554 Sci Comp & Visualization II 3.0 Credits
Covers scientific visualization, using a computational environment that includes high-performance workstations and supercomputers, and application in science and engineering. Includes applications to finite element and difference methods.
Repeat Status: Not repeatable for credit
Prerequisites: MATH 553 [Min Grade: C]
MATH 555 Topics in Sci Comp & Visualiz 3.0 Credits
Covers special topics chosen from contemporary problem areas in scientific computing and visualization, including digital image processing, wavelet transforms and their numerical treatment, numerical conformal mapping, and contemporary problem areas in scientific computing and visualization.
Repeat Status: Not repeatable for credit
Prerequisites: MATH 554 [Min Grade: C]
MATH 572 Financial Mathematics: Fixed Income Securities 3.0 Credits
The course is a mathematical introduction to interest rates and interest rates related instruments including loans, bonds, mortgages and swaps. The course emphasizes the mathematical aspects of the subject and computational implementation.
Repeat Status: Not repeatable for credit
MATH 610 Probability Theory I 3.0 Credits
Covers basics of modern probability theory: properties of probability measures, independence, Borel-Cantelli lemma, zero-one law, random variables, distribution theory, and expectations.
Repeat Status: Not repeatable for credit
Prerequisites: MATH 633 [Min Grade: C]
MATH 611 Probability Theory II 3.0 Credits
Covers further development of modern probability theory, including modes of convergence of random variables, series of random variables, weak and strong laws of large numbers, characteristics functions, inversion formula and continuity theorem, and central limit theorem.
Repeat Status: Not repeatable for credit
Prerequisites: MATH 610 [Min Grade: C]
MATH 612 Topics in Probability Theory 3.0 Credits
This third course in the probability sequence covers a selection of topics in modern probability theory. Topics may include: theory of sums of independent random variables, inequalities, martingale theory, combinatorial probability.
Repeat Status: Can be repeated 2 times for 6 credits
Prerequisites: MATH 611 [Min Grade: C]
MATH 613 Stochastic Processes I 3.0 Credits
Covers conditional probabilities, expectations, Markov chains, classification of states, recurrence and absorption probabilities, asymptotic behavior, random walk, birth and death processes, and ruin problems.
Repeat Status: Not repeatable for credit
Prerequisites: MATH 510 [Min Grade: C] and MATH 611 [Min Grade: C]
MATH 614 Stochastic Processes II 3.0 Credits
Covers queuing theory, waiting line models, embedded Markov chain method, and optimization problems. Includes applications and simulation.
Repeat Status: Not repeatable for credit
Prerequisites: MATH 613 [Min Grade: C]
MATH 615 Topics in Stochastic Processes 3.0 Credits
Covers topics including branching processes, Brownian motion, renewal processes, compounding stochastic processes, martingales, and decision-making under uncertainty.
Repeat Status: Not repeatable for credit
Prerequisites: MATH 613 [Min Grade: C]
MATH 620 Partial Differential Equations I 3.0 Credits
Covers derivation and classification of partial differential equations; elementary methods of solution, including Fourier series and transform techniques; linear and equilinear equations of the first order; hyperbolic, elliptic, and parabolic type equations; maximum principles; existence, uniqueness, and continuous dependence theorems; Riemann's method; method of characteristics; Green's functions; and variational and numerical methods.
Repeat Status: Not repeatable for credit
MATH 621 Partial Differential Equations II 3.0 Credits
Continues MATH 620.
Repeat Status: Not repeatable for credit
MATH 622 Partial Differential Equations III 3.0 Credits
Continues MATH 621.
Repeat Status: Not repeatable for credit
MATH 623 Ordinary Differential Equations I 3.0 Credits
Covers existence and uniqueness theorems, properties of solutions, adjoint equations, canonical forms, asymptotic behavior, phase space, method of isocline, classification of singular points, linear two-dimensional autonomous systems, non-linear systems, stability theory, Lyapunov's methods, quadratic forms, construction of Lyapunov's function, boundedness, limit sets, applications to controls, linear equations with periodic coefficients, Floquet theory, characteristic multipliers and exponents, existence of periodic solutions to weakly non-linear systems, jump phenomena, subharmonic resonance, and stability of periodic solutions.
Repeat Status: Not repeatable for credit
MATH 624 Ordinary Differential Equations II 3.0 Credits
Continues MATH 625.
Repeat Status: Not repeatable for credit
MATH 625 Ordinary Differential Equations III 3.0 Credits
Continues MATH 626.
Repeat Status: Not repeatable for credit
MATH 630 Complex Variables I 3.0 Credits
Covers Cauchy's theorem, Morera's theorem, infinite series, Taylor and Laurent explanations, residues, conformal mapping and applications, analytic continuation, and Riemann mapping theorem.
Repeat Status: Not repeatable for credit
MATH 631 Complex Variables II 3.0 Credits
Covers entire functions, Picard's theorem, series and product developments, Riemann Zeta function, and elliptic functions.
Repeat Status: Not repeatable for credit
Prerequisites: MATH 630 [Min Grade: C]
MATH 632 Topics in Complex Variables 3.0 Credits
Covers topics including global analytic functions, algebraic functions, and linear differential equations in the complex plane.
Repeat Status: Not repeatable for credit
Prerequisites: MATH 631 [Min Grade: C]
MATH 633 Real Variables I 3.0 Credits
Covers algebra of sets, topology of metric spaces, compactness, completeness, function spaces, general theory of measure, measurable functions, integration, convergence theorems, and applications to classical analysis and integration.
Repeat Status: Not repeatable for credit
MATH 634 Real Variables II 3.0 Credits
Covers Fubini's theorem, Radon-Nikodym theorem, LP-spaces, linear functionals on LP-spaces, Riesz-representation theorem, topological integration, Riesz-Markoz theorem, Luzin's theorem, basic complex functions, analytic functions, and complex-integration.
Repeat Status: Not repeatable for credit
Prerequisites: MATH 633 [Min Grade: C]
MATH 635 Real Variables III 3.0 Credits
Covers topics including differentiation theory, Fourier series and transforms, and singular integrals.
Repeat Status: Not repeatable for credit
Prerequisites: MATH 634 [Min Grade: C]
MATH 640 Functional Analysis 3.0 Credits
An introduction to abstract linear spaces, including normed linear spaces, Hilbert spaces, Banach spaces, and their duals. Fundamental theorems such as the Hahn-Banach theorem, open mapping and closed graph theorems will be covered, along with possible applications to differential and integral equations and fundamentals of distribution theory.
Repeat Status: Not repeatable for credit
Prerequisites: MATH 504 [Min Grade: C] and MATH 506 [Min Grade: C]
MATH 641 Harmonic Analysis 3.0 Credits
Covers modern techniques and applications of harmonic analysis, including Fourier series, Fourier transforms and related topics.
Repeat Status: Not repeatable for credit
Prerequisites: MATH 640 [Min Grade: C]
MATH 642 Operator Theory 3.0 Credits
An introduction to basic spectral theory of linear operators, theory of compact operators, and theory of unbounded operators.
Repeat Status: Not repeatable for credit
Prerequisites: MATH 640 [Min Grade: C]
MATH 643 Integral Equations I 3.0 Credits
Covers theory and application of linear integral equations, including the Hilbert-Schmidt theory. Introduces non-linear and singular integral equations and numerical methods.
Repeat Status: Not repeatable for credit
MATH 645 Transform Theory I 3.0 Credits
Covers selected topics from wavelet transforms, including properties; asymptotic analyses; and applications of the integral transforms of Laplace, Fourier, Mellin, and Radon.
Repeat Status: Not repeatable for credit
Prerequisites: MATH 640 [Min Grade: C]
MATH 646 Transform Theory II 3.0 Credits
Covers selected topics from wavelet transforms and applications, convolution equations, and the calculus of distributions.
Repeat Status: Not repeatable for credit
Prerequisites: MATH 640 [Min Grade: C] and MATH 645 [Min Grade: C]
MATH 660 Lie Groups and Lie Algebras I 3.0 Credits
Covers matrix groups, topological groups, locally isomorphic groups, universal covering groups, analytic manifolds, Lie groups; the Lie algebra of a Lie group, differential forms, and Lie's three theorems; analytic subgroups of a Lie group and compact Lie groups; and semisimple Lie algebras, general structure of Lie algebras, Cartan subalgebras, modules and representation, and computational techniques in representation theory.
Repeat Status: Not repeatable for credit
MATH 661 Lie Groups and Lie Algebras II 3.0 Credits
Continues MATH 660.
Repeat Status: Not repeatable for credit
MATH 662 Lie Groups/Algebras III 3.0 Credits
Continues MATH 661.
Repeat Status: Not repeatable for credit
MATH 670 Methods of Optimization I 3.0 Credits
Provides a rigorous treatment of theory and computational techniques in linear programming and its extensions, including formulation, duality theory, simplex and dual-simplex methods, and sensitivity analysis; network flow problems and algorithms; systems of inequalities, including exploiting special structure in the simplex method and use of matrix decompositions; and applications in game theory and integer programming.
Repeat Status: Not repeatable for credit
MATH 671 Methods of Optimization II 3.0 Credits
Covers necessary and sufficient conditions for unconstrained and constrained optimization. Includes computational methods including quasi-Newtonian and successive quadratic programming, and penalty and interior methods.
Repeat Status: Not repeatable for credit
MATH 672 Methods of Optimization III 3.0 Credits
Covers advanced topics in mathematical programming, including interior point methods in linear programming; stochastic optimization; multi-objective optimization; and global minimax, functional, and non-linear least squares optimization methods.
Repeat Status: Not repeatable for credit
Prerequisites: MATH 670 [Min Grade: C] and MATH 671 [Min Grade: C]
MATH 673 Calculus of Variations 3.0 Credits
Introduction to calculus of variations. Covers applications to geometry, classical mechanics and control theory, Euler-Lagrange equations, problems with constraints, canonical equations, Hamiltonian mechanics, symmetries and Noether's theorem, Hamilton-Jacobi theory, introduction to optimal control, maximum principle, and Hamilton-Jacobi-Bellman equations.
Repeat Status: Not repeatable for credit
MATH 680 Special Topics 0.5-9.0 Credits
Covers special topics of interest to students and faculty.
Repeat Status: Can be repeated multiple times for credit
MATH 699 Independent Study in Math 0.5-6.0 Credits
College/Department: College of Arts and Sciences
Repeat Status: Can be repeated multiple times for credit
MATH 701 Algebraic Combinatorics 3.0 Credits
This course covers methods of Abstract Algebra that can be applied to various combinatorial problems and conversely, combinatorial methods to approach problems in representation theory, algebraic geometry, and homological algebra.
Repeat Status: Not repeatable for credit
Prerequisites: MATH 533 [Min Grade: C] and MATH 534 [Min Grade: C]
MATH 723 Mathematical Neuroscience 3.0 Credits
This is an introduction to mathematical and computational techniques for analyzing neuronal models. Topics include conductance based models, neuronal excitability, bursting, neural networks, and compartmental models, as well as phase plane analysis, slow-fast systems, elements of applied bifurcation theory, and simulating differential equation models using MATLAB.
Repeat Status: Not repeatable for credit
MATH 799 Independent Study in Math 6.0 Credits
College/Department: College of Arts and Sciences
Repeat Status: Can be repeated multiple times for credit
MATH 898 Master's Thesis 0.5-20.0 Credits
Master's thesis.
Repeat Status: Not repeatable for credit
MATH 997 Research 1.0-12.0 Credit
Research.
Repeat Status: Can be repeated multiple times for credit
MATH 998 Ph.D. Dissertation 1.0-12.0 Credit
Ph.D. dissertation.
Repeat Status: Can be repeated multiple times for credit
